These equivalences remain all valid if quantification is done on more than one variable. The latter explains how you can reduce the universal quantification to the existential one, and for the existential one, you know how to do it, right?

And a direct answer to your question about replacing OR with AND in the call to APPLY in the algorithm: Yes, that is what will happen there, if you will directly implement Forall as a BDD algorithm.

You only do one of the two, i.e., either use Exists with $\neg\varphi$ and negate its result, or use a direct implementation of the Forall algorithm that you obtain essentially by replacing the OR by AND. But I would have to check is further changes are done in Forall (e.g. in the base cases).